Question

The Cartesian equation of the line passing through the points A(2, 2, 1) and B(1, 3, 0) is

• A. $\frac{x+2}{1} = \frac{y+2}{-1} = \frac{z+1}{-1}$
• B. $\frac{x-2}{-1} = \frac{y-2}{1} = \frac{z-1}{-1}$
• C. $\frac{x+2}{-1} = \frac{y+2}{1} = \frac{z+1}{-1}$
• D. None of these

Hint:

Always double-check your direction ratios by subtracting the coordinates in the same order. Using point B as the reference point instead of A would give an equivalent equation, but checking the options reveals that point A was used in the numerators.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find the 3D Cartesian equation of a straight line that passes through two given coordinate points.

Step 2: Key Formula or Approach:
The Cartesian equation of a line passing through two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by the formula:
$$\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$$

Step 3: Detailed Explanation:
The given points are $A(2, 2, 1)$ and $B(1, 3, 0)$.
Let $(x_1, y_1, z_1) = (2, 2, 1)$ and $(x_2, y_2, z_2) = (1, 3, 0)$.
Calculate the direction ratios (the denominators):
$$x_2 - x_1 = 1 - 2 = -1$$ $$y_2 - y_1 = 3 - 2 = 1$$ $$z_2 - z_1 = 0 - 1 = -1$$ Substitute the point $A$ and the direction ratios into the formula:
$$\frac{x - 2}{-1} = \frac{y - 2}{1} = \frac{z - 1}{-1}$$

Step 4: Final Answer:
The Cartesian equation is $\frac{x-2}{-1} = \frac{y-2}{1} = \frac{z-1}{-1}$, matching option (B).